Noncommutative stationary processes / Rolf Gohm

This book deals with completely positive maps on C∗-algebras and von Neumann algebras, and their dilations thought of as noncommutative generalizations of stochastic maps and classical Markov processes. The first chapter of the book is on a duality between extensions and dilations and an idea about this can be obtained from the review [R. Gohm, in Advances in quantum dynamics (South Hadley, MA, 2002), 139–147, Contemp. Math., 335, Amer. Math. Soc., Providence, RI, 2003; MR2026016 (2005b:46133)]. The second chapter compares some of the different notions of Markov processes available in the literature. Detailed comparisons are made between approaches of B. Kümmerer [in Quantum probability communications, Vol. XI (Grenoble, 1998), 273–304, World Sci. Publishing, River Edge, NJ, 2003; MR2032370] and B. V. R. Bhat and K. R. Parthasarathy [Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 4, 601–651; MR1355610 (96i:46079)]. A brief account of Kümmerer-Maassen scattering theory is also given. The third chapter takes a very general axiomatic approach for adaptedness of noncommutative processes leading to a definition of what the author calls adapted endomorphisms. Some specific instances where these ideas can be useful are presented. Finally in the last chapter a host of examples and applications are presented. This neat little book is very readable and does not require much preliminaries. Most of the examples are quite elementary and need just matrix theory. As such, it should be useful for those who wish to get introduced into this growing area of quantum probability. Of course, being a research monograph it is also of interest to researchers in the field. (MathSciNet)